Method and system for multi-period performance attribution with metric-preserving coefficients

ABSTRACT

A method for arithmetic performance attribution which accurately links single-period attribution effects over multiple periods. In preferred embodiments, the method determines portfolio relative performance over multiple time periods (t=1, 2, . . . , T) as a sum of terms of form  
           R   -     R   _       =       ∑     i                 t            ⌊         c   1          a     i                 t         +       c   2          a     i                 t     2         ⌋         ,                 
 
     where a it  is a component of active return for period t, the summation over index i is a summation over all components a it  for period t, R is  
         R   =       [       ∏     t   =   1     T          (     1   +     R   t       )       ]     -   1       ,                  R   _     =       [       ∏     t   =   1     T          (     1   +       R   _     t       )       ]     -   1       ,                 
 
     R t  is a portfolio return for period t, {overscore (R)} t  is a benchmark return for period t, and the coefficients c 1  and c 2  are c 1 =A, and  
         c   2     =       [       R   -     R   _     -     A          ∑     j                 t            a     j                 t                 ∑     j                 t            a     j                 t     2         ]     .                   
 
     More generally, the invention is an arithmetic method for determining portfolio relative performance over multiple time periods (t=1, 2, . . . , T) as a sum of terms of form:  
           R   -     R   _       =       ∑     i                 t              ∑     k   =   1     ∞            c   k          a     i                 t     k             ,                 
 
     where a it  is a component of active return for period t. In preferred quadratic implementations (in which the only nonzero coefficients c k  are those for which k=1 or k=2), the coefficients c 1  and c 2  are defined as in the above-mentioned preferred embodiments. In all embodiments, the method of the invention is metric preserving at the component portfolio level. Other aspects of the invention are a computer system programmed to perform any embodiment of the inventive method, and a computer readable medium which stores code for implementing any embodiment of the inventive method.

CROSS-REFERENCE TO RELATED APPLICATION

[0001] The present application is a continuation-in-part of pending U.S.application Ser. No. 09/698,693, filed on Oct. 27, 2000 (assigned to theassignee of the present application), and a continuation-in-part ofpending U.S. application Ser. No. 09/613,855, filed on Jul. 11, 2000(assigned to the assignee of the present application).

TECHNICAL FIELD OF THE INVENTION

[0002] The present invention relates to methods for performingperformance attribution to compare the returns of a financial portfolioagainst those of a benchmark, and attribute the relative performance tovarious effects resulting from active decisions by the portfoliomanager. More particularly, the invention is an improved method forlinking single-period attribution effects over multiple periods, usingan arithmetic methodology.

BACKGROUND OF THE INVENTION

[0003] In performing performance attribution, the returns of a portfolioare compared against those of a benchmark, and the excess return (i.e.,relative performance) is attributed to various effects resulting fromactive decisions by the portfolio managers. Performance attribution is arich and complex topic, which can be viewed from many angles. There area variety of conventional methods for performing attribution based on asingle-period analysis. However, if performance is measured over anextended length of time, a single-period buy-and-hold analysis may leadto significant errors, especially for highly active portfolios.Therefore, it is imperative to link the single-period attributioneffects over multiple periods in an accurate and meaningful way. The twobasic approaches that have arisen for such linking are the arithmeticand geometric methodologies.

[0004] In arithmetic attribution, the performance of a portfoliorelative to a benchmark is given by the difference R−{overscore (R)},where R and {overscore (R)} refer to portfolio and benchmark returns,respectively. This relative performance, in turn, is decomposed sectorby sector into attribution effects that measure how well the portfoliomanager weighted the appropriate sectors and selected securities withinthe sectors. The sum of the attribution effects gives the performance,R−{overscore (R)}.

[0005] In geometric attribution, by contrast, the relative performanceis defined by the ratio (1+R)/(1+{overscore (R)}). This relativeperformance is again decomposed sector by sector into attributioneffects. In this case, however, it is the product of the attributioneffects that gives the relative performance (1+R) / (1+{overscore (R)}).A recent example of both arithmetic and geometric attribution systems isdescribed in Carino, “Combining Attribution Effects Over Time,” Journalof performance Measurement, Summer 1999, pp. 5-14 (“Carino”).

[0006] An advantage of the arithmetic approach is that it is moreintuitive. For instance, if the portfolio return is 21% and thebenchmark return is 10%, most people regard the relative performance tobe 11%, as opposed to 10%. An advantage of geometric attribution, on theother hand, is the ease with which attribution effects can be linkedover time.

[0007] Carino describes one possible algorithm for linking attributioneffects over time that results in a multi-period arithmetic performanceattribution system. Furthermore, the result is residual free in that thesum of the linked attribution effects is exactly equal to the differencein linked returns. Carino discloses an arithmetic performanceattribution method which determines portfolio relative performance overmultiple time periods as a sum of terms of form (R_(t)−{overscore(R)}_(t))β_(t), where the index “t” indicates one time period, and whereCarino's coefficients β_(t) are$\beta_{t}^{Carino} = {\left\lbrack \frac{R - \overset{\_}{R}}{{\ln \left( {1 + R} \right)} - {\ln \left( {1 + \overset{\_}{R}} \right)}} \right\rbrack {\left( \frac{{\ln \left( {1 + R_{t}} \right)} - {\ln \left( {1 + {\overset{\_}{R}}_{t}} \right)}}{R_{t} - {\overset{\_}{R}}_{t}} \right).}}$

[0008] In accordance with the present invention, new coefficients to bedefined below replace Carino's coefficients β_(t) (sometimes referred toherein as conventional “logarithmic” coefficients). The inventivecoefficients have a much smaller standard deviation than theconventional logarithmic coefficients and are metric preserving.Reducing the standard deviation of the coefficients is important inorder to minimize the distortion that arises from overweighting certainperiods relative to others.

SUMMARY OF THE INVENTION

[0009] In a class of preferred embodiments, the invention is anarithmetic method for determining portfolio relative performance overmultiple time periods (t=1, 2, . . . ,T) as a sum of terms of form${{R - \overset{\_}{R}} = {\sum\limits_{i\quad t}\left\lfloor {{c_{1}a_{i\quad t}} + {c_{2}a_{i\quad t}^{2}}} \right\rfloor}},{\text{where}\text{R}\text{is}}$${R = {\left\lbrack {\prod\limits_{t = 1}^{T}\left( {1 + R_{t}} \right)} \right\rbrack - 1}},{\overset{\_}{R}\quad {is}}$${\overset{\_}{R} = {\left\lbrack {\prod\limits_{t = 1}^{T}\left( {1 + {\overset{\_}{R}}_{t}} \right)} \right\rbrack - 1}},$

[0010] R_(t) is a portfolio return for period t, {overscore (R)}_(t) isa benchmark return for period t, a_(it) is a component of active return(e.g., issue selection for a given sector) for period t, the componentsa_(it) for each period t satisfy${{\sum\limits_{i}\quad a_{it}} = {R_{t} - {\overset{\_}{R}}_{t}}},$

[0011] the summation over index i is a summation over all the componentsa_(it) for period t, the summation over index t is a summation over thetime periods, and the coefficients c₁ and c₂ are respectively c₁=A and${c_{2} = \left\lbrack \frac{R - \overset{\_}{R} - {A{\sum\limits_{j\quad t}a_{j\quad t}}}}{\sum\limits_{j\quad t}a_{j\quad t}^{2}} \right\rbrack},$

[0012] where the summation over index j is a summation over all thecomponents a_(it) for period t.

[0013] The inventive method is metric preserving at the component level.Preferably, the value A is given by${A = {{\frac{1}{T}\left\lbrack \frac{\left( {R - \overset{\_}{R}} \right)}{\left( {1 + R} \right)^{1/T} - \left( {1 + \overset{\_}{R}} \right)^{1/T}} \right\rbrack}\quad {for}\quad \left( {R \neq \overset{\_}{R}} \right)}},{and}$$A = {\left( {1 + R} \right)^{{({T - 1})}/T}\quad {for}\quad {\left( {R = \overset{\_}{R}} \right).}}$

[0014] More generally, the invention is an arithmetic method fordetermining portfolio relative performance over multiple time periods(t=1, 2, . . . ,T) as a sum of terms of form:${{R - \overset{\_}{R}} = {\sum\limits_{i\quad t}{\sum\limits_{k = 1}^{\infty}{c_{k}a_{i\quad t}^{k}}}}},$

[0015] where a_(it) is a component of active return for period t. In thepreferred quadratic case (in which the only nonzero coefficients c_(k)are those for which k=1 or k=2), the coefficients c₁ and c₂ are definedas in the above-mentioned preferred embodiments. In all embodiments, theinventive method is metric preserving at the component level.

[0016] Other aspects of the invention are a computer system programmedto perform any embodiment of the inventive method, and a computerreadable medium which stores code for implementing any embodiment of theinventive method.

BRIEF DESCRIPTION OF THE DRAWINGS

[0017]FIG. 1a is a contour plot of the average logarithmic coefficients,determined in accordance with the prior art, resulting from a set ofsimulations.

[0018]FIG. 1b is a contour plot of average metric preservingcoefficients (defined herein) resulting from the same simulations whichdetermined FIG. 1 a.

[0019]FIG. 2a is plot of normalized standard deviation for theconventional logarithmic coefficients, assuming the same set ofdistributions that were assumed to generate FIGS. 1a and 1 b.

[0020]FIG. 2b is plot of normalized standard deviation for a set ofmetric preserving coefficients (defined herein), assuming the same setof distributions that were assumed to generate FIGS. 1a and 1 b.

[0021]FIG. 3 is a block diagram of a computer system for implementingany embodiment of the inventive method.

[0022]FIG. 4 is an elevational view of a computer readable optical diskon which is stored computer code for implementing any embodiment of theinventive method.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0023] The arithmetic performance attribution method of the presentinvention is an improved approach to arithmetic linking over multipleperiods. Preferred embodiments of the invention are based on an optimaldistribution of the residual among the different time periods. Such anapproach minimizes the distortion that arises from overweighting certaintime periods relative to others. The resulting attribution system isalso residual free, robust, and completely general, so that performancecan be linked without complication for any set of sector weights andreturns.

Single-Period Arithmetic Attribution

[0024] The portfolio return R_(t) for a single period t can be writtenas the weighted average return over N sectors $\begin{matrix}{{R_{t} = {\sum\limits_{i = 1}^{N}{w_{i\quad t}r_{i\quad t}}}},} & (1)\end{matrix}$

[0025] where w_(it) and r_(it) are the portfolio weights and returns forsector i and period t, respectively. For the benchmark, thecorresponding returns are $\begin{matrix}{{{\overset{\_}{R}}_{t} = {\sum\limits_{i = 1}^{N}\quad {{\overset{\_}{w}}_{it}{\overset{\_}{r}}_{it}}}},} & (2)\end{matrix}$

[0026] with the overbar denoting the benchmark. The arithmetic measureof relative performance is therefore $\begin{matrix}{{R_{t} - {\overset{\_}{R}}_{t}} = {{\sum\limits_{i = 1}^{N}\quad {w_{it}r_{it}}} - {\sum\limits_{i = 1}^{N}\quad {{\overset{\_}{w}}_{it}{{\overset{\_}{r}}_{it}.}}}}} & (3)\end{matrix}$

[0027] This difference can be rewritten as $\begin{matrix}{{R_{t} - {\overset{\_}{R}}_{t}} = {{\sum\limits_{i = 1}^{N}\quad {w_{it}r_{it}}} - {\sum\limits_{i = 1}^{N}\quad {{\overset{\_}{w}}_{it}{\overset{\_}{r}}_{it}}} + {\quad{{\left\lbrack {{\sum\limits_{i = 1}^{N}\quad {w_{ir}{\overset{\_}{r}}_{it}}} - {\sum\limits_{i = 1}^{N}\quad {w_{it}{\overset{\_}{r}}_{it}}}} \right\rbrack + \left\lbrack {{\sum\limits_{i = 1}^{N}\quad {{\overset{\_}{w}}_{it}{\overset{\_}{R}}_{t}}} - {\sum\limits_{i = 1}^{N}\quad {w_{it}{\overset{\_}{R}}_{t}}}} \right\rbrack},}}}} & (4)\end{matrix}$

[0028] by noting that the terms in brackets are equal to zero. Combiningterms, we obtain the desired result $\begin{matrix}{{R_{t} - {\overset{\_}{R}}_{t}} = {{\sum\limits_{i = 1}^{N}\quad {w_{it}\left( {r_{it} - {\overset{\_}{r}}_{it}} \right)}} + {\sum\limits_{i = 1}^{N}\quad {\left( {w_{it} - {\overset{\_}{w}}_{it}} \right){\left( {{\overset{\_}{r}}_{it} - {\overset{\_}{R}}_{t}} \right).}}}}} & (5)\end{matrix}$

[0029] We interpret the terms in the first summation to be the issueselection

I _(it) ^(A) =w _(it) (r _(it) −{overscore (r)} _(it)),   (6)

[0030] with the superscript A denoting arithmetic.

[0031] The issue selection I_(it) ^(A) measures how well the portfoliomanager picked overperforming securities in sector i during period t.

[0032] Similarly, the terms in the second summation of equation (5) weinterpret to be the sector selection,

S _(it) ^(A)=(w _(it) ={overscore (W)} _(it))({overscore (R)} _(it)−{overscore (R)} _(t))  (7)

[0033] which measures the extent to which the manager overweighted theout-performing sectors. The active contribution A_(it) ^(A) is the sumof the issue selection I_(it) ^(A) and sector selection S_(it) ^(A):

[0034] and gives the contribution of sector i to the performance forperiod t due to active management decisions.

[0035] The above relations allow us to write the net performance forperiod t as $\begin{matrix}{{R_{t} - {\overset{\_}{R}}_{t}} = {{\sum\limits_{i = 1}^{N}\quad \left( {I_{it}^{A} + S_{it}^{A}} \right)} = {\sum\limits_{i = 1}^{N}\quad {A_{it}^{A}.}}}} & (9)\end{matrix}$

[0036] To summarize, the single-period relative performance has beenfully decomposed into attribution effects at the sector level. Theseattribution effects, when summed over all sectors, give the total excessreturn for the period, R_(t)−{overscore (R)}_(t).

Multiple-Period Arithmetic Attribution

[0037] It is desirable to extend the above analysis to themultiple-period case. The portfolio and benchmark returns linked over Tperiods are respectively given by $\begin{matrix}{{{1 + R} = {\prod\limits_{t = 1}^{T}\quad \left( {1 + R_{t}} \right)}},{{1 + \overset{\_}{R}} = {\prod\limits_{t = 1}^{T}\quad {\left( {1 + {\overset{\_}{R}}_{t}} \right).}}}} & (10)\end{matrix}$

[0038] Just as we define the relative performance for the single-periodcase by the difference in single-period returns, it is natural to definethe relative performance for the multiple-period case as the differencein linked returns, R−{overscore (R)}.

[0039] If the returns are small, then the relative performance isapproximately given by $\begin{matrix}{{R - \overset{\_}{R}} \approx {\sum\limits_{t = 1}^{T}\quad {\left( {R_{t} - {\overset{\_}{R}}_{t}} \right).}}} & (11)\end{matrix}$

[0040] However, this approximation breaks down for large returns. Abetter approach is to multiply the right side of (11) by a constantfactor A that takes into account the characteristic scaling which arisesfrom geometric compounding: $\begin{matrix}{{R - \overset{\_}{R}} \approx {A{\sum\limits_{t = 1}^{T}\quad {\left( {R_{t} - {\overset{\_}{R}}_{t}} \right).}}}} & (12)\end{matrix}$

[0041] An obvious possible choice for A is given by $\begin{matrix}{\frac{R - \overset{\_}{R}}{\sum\limits_{t = 1}^{T}\quad \left( {R_{t} - {\overset{\_}{R}}_{t}} \right)}.} & (13)\end{matrix}$

[0042] However, this naive solution is unacceptable because it does notnecessarily reflect the characteristic scaling of the system.Furthermore, it may easily occur that the numerator and denominator ofthe above expression have opposite sign, in which case the entirelinking process loses its underlying meaning.

[0043] The value of A that correctly describes such scaling can be foundby substituting the mean geometric return (1+R)^(1/T−1) for thesingle-period returns R_(t), and similarly for the benchmark. Therefore,in preferred embodiments, A is given by $\begin{matrix}{{A = {\frac{1}{T}\left\lbrack \frac{\left( {R - \overset{\_}{R}} \right)}{\left( {1 + R} \right)^{1/T} - \left( {1 + \overset{\_}{R}} \right)^{1/T}} \right\rbrack}},\quad {\left( {R \neq \overset{\_}{R}} \right).}} & (14)\end{matrix}$

[0044] Note that A satisfies the required property of being alwayspositive. For the special case R={overscore (R)}, it is easy to showthat the above expression has limiting value

A=(1+R)^((T−1)/T), (R={overscore (R)}).  (15)

[0045] In alternative embodiments, A is taken to have some other value.For example, A=1 or A=[(1+R)(1+{overscore (R)})]^(½) in alternativeembodiments.

[0046] Although (12) is a good approximation with A defined by equations(14) and (15), it still leaves a small residual for general sets ofreturns. However, we can introduce a set of corrective terms α_(t) atthat distribute the residual among the different periods so that thefollowing equation exactly holds $\begin{matrix}{{R - \overset{\_}{R}} = {\sum\limits_{t = 1}^{T}{\left( {A + \alpha_{t}} \right){\left( {R_{t} - {\overset{\_}{R}}_{t}} \right).}}}} & (16)\end{matrix}$

[0047] The problem now reduces to calculating the α_(t). Our objectiveis to construct a solution for equation (16) that minimizes thedistortion arising from overweighting certain periods relative toothers. In other words, the α_(t) should be chosen to be as small aspossible. In order to find the optimal solution, we must minimize thefunction $\begin{matrix}{{f = {\sum\limits_{t = 1}^{T}\alpha_{t}^{2}}},} & (17)\end{matrix}$

[0048] subject to the constraint of equation (16). This is a standardproblem involving Lagrange multipliers, and the optimal solution isgiven by $\begin{matrix}{\alpha_{t} = {\left\lbrack \frac{R - \overset{\_}{R} - {A{\sum\limits_{k = 1}^{T}\left( {R_{k} - {\overset{\_}{R}}_{k}} \right)}}}{\sum\limits_{k = 1}^{T}\left( {R_{k} - {\overset{\_}{R}}_{k}} \right)^{2}} \right\rbrack {\left( {R_{t} - {\overset{\_}{R}}_{t}} \right).}}} & (18)\end{matrix}$

[0049] With the α₁ thus determined, the linking problem is solved. Theoptimized linking coefficients, denoted β_(t) ^(Vestek), are thus givenby

β_(t) ^(Vestek) =A+α _(t),  (19)

[0050] with A defined in equations (14) and (15), and α_(t) given byequation (18). Substituting equation (9) and equation (19) into equation(16) we obtain $\begin{matrix}{{R - \overset{\_}{R}} = {\sum\limits_{t = 1}^{T}{\sum\limits_{i = 1}^{N}{{\beta_{t}^{Vestek}\left( {I_{i\quad t}^{A} + S_{i\quad t}^{A}} \right)}.}}}} & (20)\end{matrix}$

[0051] Observe that this result is fully additive, so that the totalperformance is defined as a sum of attribution effects, each summed oversectors and time periods. Furthermore, there is no unexplained residual.

[0052] The inventor has determined that if one chooses the value of A tobe the value determined by equation (14) (or equation (15), ifR={overscore (R)}), the standard deviation of the optimized coefficientsof equation (19) is less than that for the logarithmic coefficientsdisclosed in the above-cited paper by Carino, namely the β_(t) ^(Carino)of equation (21), in all simulations performed. Thus, this choice forthe value of A guarantees smaller standard deviation among thecoefficients β_(t) ^(Vestek) than among the logarithmic coefficientstaught by Carino.

[0053] It is interesting to compare the optimized weighting coefficientsβ_(t) ^(Vestek) of equation (19) to the logarithmic coefficientsdisclosed by Carino: $\begin{matrix}{\beta_{t}^{Carino} = {\left\lbrack \frac{R - \overset{\_}{R}}{{\ln \left( {1 + R} \right)} - {\ln \left( {1 + \overset{\_}{R}} \right)}} \right\rbrack {\left( \frac{{\ln \left( {1 + R_{t}} \right)} - {\ln \left( {1 + {\overset{\_}{R}}_{t}} \right)}}{R_{t} - {\overset{\_}{R}}_{t}} \right).}}} & (21)\end{matrix}$

[0054] The logarithmic coefficients (21) are similar to their optimizedcounterparts (19) in that both lead to residual-free linking. However,the logarithmic coefficients tend to overweight periods withlower-than-average returns, and to underweight those withhigher-than-average returns. This appears to be an artifact of thelinking algorithm, and not to be grounded in any economic principle. Theoptimized coefficients, by contrast, tend to weight each period asevenly as possible.

[0055] The inventor has conducted a more detailed analysis comparing theoptimized coefficients and the conventional logarithmic coefficients,using computational simulations linking single-month attribution effectsover a twelve-month period. The portfolio and benchmark returns weredrawn from normal distributions, with the standard deviation set equalto the absolute value of the mean return. The portfolio and benchmarkdistributions were kept fixed for the twelve-month period, and each datapoint was calculated by averaging the linking coefficients over 1000sample paths drawn from the same fixed distributions. The mean monthlyreturns were then varied from −10% to +20%, in order to obtain anunderstanding of the global behavior of the linking coefficients.Typical annual returns varied from −70% on the low end to +800% on thehigh end. FIGS. 1a and 1 b show results of the simulations, with FIG. 1abeing a contour plot of the average logarithmic coefficients and FIG. 1bbeing a contour plot of the average optimized coefficients. In bothcases, the coefficients increase from an average of less than 0.5 forthe smallest returns to more than 6.0 for the largest returns.Furthermore, we see that for any combination of portfolio and benchmarkreturns, the average coefficient is virtually identical in bothapproaches. Evidently, the reason for this similarity is that thecoefficients in the logarithmic algorithm also correctly account for thescaling properties.

[0056] A more interesting study, however, is to compare the standarddeviation for both sets of coefficients for the same set of returns usedin FIGS. 1a and 1 b. We first calculate for a single twelve-month period{circumflex over (σ)}, the percent standard deviation of the linkingcoefficients normalized by the average linking coefficient <β> for thattwelve-month period, $\begin{matrix}{\hat{\sigma} = {100{\frac{\sqrt{{\langle\beta^{2}\rangle} - {\langle\beta\rangle}^{2}}}{\langle\beta\rangle}.}}} & (22)\end{matrix}$

[0057] We then average {circumflex over (σ)} over 1000 sample paths inorder to obtain a good estimate of the average normalized standarddeviation of the linking coefficients. The resulting contour plots arepresented in FIGS. 2a and 2 b. We observe fundamentally distinctbehavior for the two cases. For the logarithmic coefficients, thenormalized standard deviation increases in concentric circles about theorigin, rising to over 10% for the largest returns considered here. Bycontrast, the optimized coefficients exhibit valleys of extremely lowstandard deviation extending along the directions R=±{overscore (R)}.This property of the optimized coefficients is very appealing because,in the usual case, portfolio returns can be expected to at least roughlytrack the benchmark returns. In other words, in the usual case, theoptimized coefficients have a much smaller standard deviation than theconventional logarithmic coefficients.

[0058] Although the results of FIGS. 1a, 1 b, 2 a, and 2 b were obtainedfor a twelve-month period with specific distributions, the inventor hasconducted extensive simulations with different periods and differentdistributions and has found that the results are entirely consistentwith those shown in FIGS. 1a, 1 b, 2 a, and 2 b.

[0059] It is natural to ask what kinds of differences might arise inpractice between the two sets of linking coefficients. In Table 1 wepresent a hypothetical set of portfolio and benchmark returns for asix-month period, together with the resulting linking coefficients forthe logarithmic and optimized cases. We note that the standard deviationof the optimized coefficients is very small, with the coefficientsranging from roughly 1.41 to 1.42. For the logarithmic case, on theother hand, the coefficients range from 1.26 to 1.54. The linkedportfolio and benchmark returns for this example are 64.37% and 39.31%,respectively, for an excess return of 25.06%. In Table 1 we alsodecompose the single-period relative performance into issue selectionI_(t) ^(A) and sector selection S_(t) ^(A), where these attributioneffects represent the total summed over all sectors. The values werespecifically chosen for illustrative purposes with the averagesingle-period issue selection and sector selection being equal. Applyingthe logarithmic linking algorithm, we find that the linked issueselection is 10.88%, and that the linked sector selection is 14.18%.Using the optimized coefficients, the corresponding values are 12.52%and 12.54%, respectively. In both cases, the issue selection and sectorselection add to give the correct relative performance of 25.06%, sothat there is no residual in either method. However, the optimizedapproach more accurately reflects the fact that, on average, the issueselection and sector selection were equal. Table 1 Comparison of thelogarithmic (β_(t) ^(Carlino)) and optimized (β_(t) ^(Vestek))coefficients for a hypothetical six-month period. Portfolio andbenchmark returns are given by R_(t) and {overscore (R)}_(t),respectively. Also presented are the single-period issue selection I_(t)^(A) and sector selection S_(t) ^(A). Period t R_(t) (%) {overscore(R)}_(t) (%) β_(t) ^(Carino) β_(t) ^(Vestek) I_(t) ^(A) (%) S_(t) ^(A)(%) 1 10.0 5.0 1.409496 1.412218 2.0 3.0 2 25.0 15.0 1.263177 1.4106069.0 1.0 3 10.0 20.0 1.318166 1.417053 −2.0 −8.0 4 −10.0 10.0 1.5200151.420276 −13.0 −7.0 5 5.0 −8.0 1.540243 1.409639 3.0 10.0 6 15.0 −5.01.447181 1.407383 10.0 10.0

[0060] A major advantage of the above-discussed optimized coefficientsβ_(t) ^(Vestek=)A+α_(t) of equation (16), with α_(t) defined in equation(14) or (15) and a, defined in equation (18), over the logarithmiccoefficients of equation (21) is that the coefficients β_(t)^(Vestek)=A+α_(t) are metric preserving at the portfolio level, wherethe expression “metric preserving” at the portfolio level is used in thefollowing sense. For simplicity, we will delete the superscript “Vestek”when referring below to the coefficients β_(t) ^(Vestek). The statementthat a coefficient β_(t) is “metric preserving” at the portfolio leveldenotes herein that, for any two periods (t1 and t2) having the samesingle period active return (R_(t1)−{overscore(R)}_(t1))=(R_(t2)−{overscore (R)}_(t2)), the quantities fitβ_(t)(R_(t)−{overscore (R)}_(t)) in the total portfolio performance(linked over all time periods) satisfy β_(f1)(R_(t1)−{overscore(R)}_(f1))=β_(t2)(R_(t2)−{overscore (R)}_(t2)). Expressed another way,using the optimized coefficients, it is possible to write the activereturn for the linked case as $\begin{matrix}{{{R - \overset{\_}{R}} = {\sum\limits_{t = 1}^{T}\left\lbrack {{c_{1}\left( {R_{t} - {\overset{\_}{R}}_{t}} \right)} + {c_{2}\left( {R_{t} - {\overset{\_}{R}}_{t}} \right)}^{2}} \right\rbrack}},} & (23)\end{matrix}$

[0061] where c₁=A and c₂ is given by $\begin{matrix}{c_{2} = {\left\lbrack \frac{R - \overset{\_}{R} - {A{\sum\limits_{j = 1}^{T}\left( {R_{j} - {\overset{\_}{R}}_{j}} \right)}}}{\sum\limits_{j = 1}^{T}\left( {R_{j} - {\overset{\_}{R}}_{j}} \right)^{2}} \right\rbrack.}} & (24)\end{matrix}$

[0062] Since c₁ and c₂ are independent of t, it is clear from equation(23) that two periods which have the same active return will alsocontribute equally to the active return in the linked case (i.e., themethodology is metric preserving at the portfolio level). In contrast,using the logarithmic coefficients, it is impossible to express theactive return for the linked case in the form given by equation (23),and hence the methodology is not metric preserving at the portfoliolevel.

[0063] Before defining a class of embodiments of the invention whichemploy metric preserving coefficients, we note that although theabove-described algorithm employing the optimized coefficients ofequation (16) with A defined as in equation (14) or (15) and α_(t)defined as in equation (18) (the “optimized algorithm”) is metricpreserving at the portfolio level (so that two periods with the samesingle period active return contribute the same amount to the activereturn in the multiple period case), this does not mean that theoptimized algorithm is metric preserving at the component level. Toappreciate the latter statement, let a_(it) be a component of activereturn (e.g., issue selection for a given sector) for period t, and leta_(i′t′) be another component of active return (for another period t′).Now suppose that a_(it)=a_(i′t′), so that the two components contributethe same amount to the relative performance of the single periods. Inthe linked case, the two attribution effects will contribute β_(t)a_(it)and β_(t′)a_(i′t′), respectively. Although the optimized coefficientsare constructed so that β_(t)≈β_(t′), in general these two linkingcoefficients will not be identical (unless, of course, the activereturns for periods t and t′ are the same). Therefore, in general, theoptimized algorithm is not strictly metric preserving at the componentlevel (where “metric preserving at the component level” denotes that,for any two components a_(it) and a_(i′t′), that satisfya_(it)=a_(i′t′), the two components will contribute equal amountsβ_(t)a_(it)=β_(t′)a_(i′t′) to the linked portfolio performance over alltime periods).

[0064] In a class of preferred embodiments, the inventive method is amodification of the optimized algorithm, which is metric preserving atthe component level. We assume that the components a_(it) for eachperiod must add to give the active return for the period,$\begin{matrix}{{\sum\limits_{i}a_{i\quad t}} = {R_{t} - {{\overset{\_}{R}}_{t}.}}} & (25)\end{matrix}$

[0065] Equation (23) is modified and generalized in accordance with theinvention to yield $\begin{matrix}{{{R - \overset{\_}{R}} = {\sum\limits_{i\quad t}\left\lfloor {{c_{1}a_{i\quad t}} + {c_{2}a_{i\quad t}^{2}}} \right\rfloor}},} & (26)\end{matrix}$

[0066] where, as before, c₁=A, but now c₂ is given by $\begin{matrix}{c_{2} = {\left\lbrack \frac{R - \overset{\_}{R} - {A{\sum\limits_{j\quad t}a_{j\quad t}}}}{\sum\limits_{j\quad t}a_{j\quad t}^{2}} \right\rbrack.}} & (27)\end{matrix}$

[0067] Because c₁ and c₂ are independent of period (t) and componentindex (i), it is clear that equation (26) is metric preserving at thecomponent level. It is within the scope of the invention to determineportfolio performance relative to a benchmark (over multiple timeperiods t, where t varies from 1 to 7) in accordance with equation (26),with c₁=A, and c₂ determined by equation (27). In preferred ones of suchembodiments, the value of A is given by equation (14) for R≠{overscore(R)} and by equation (15) for the special case that R={overscore (R)}.In other embodiments in this class, A is taken to have some other value.For example, A=1 or A=[(1+R)(1+{overscore (R)})]^(½) in some alternativeembodiments. Note that if we consider the attribution effect to be theentire active return, then equations (26) and (27) reduce to thefamiliar results given by equations (23) and (24).

[0068] In a more broadly defined class of embodiments of the invention,portfolio performance is determined in accordance with the followinggeneralized version of equation (26), which includes higher order terms(e.g., cubics, quartics, and so on): $\begin{matrix}{{R - \overset{\_}{R}} = {\sum\limits_{i\quad t}{\sum\limits_{k = 1}^{\infty}{c_{k}{a_{i\quad t}^{k}.}}}}} & (28)\end{matrix}$

[0069] While equation (28) represents the most general expression forthe metric preserving embodiments of the linking methods of theinvention, preferred implementations include nonzero coefficients onlyfor the first two terms (k=1 and k=2), and c₁ and c₂ are as defined inthe above-described class of embodiments. In implementations thatinclude terms beyond the quadratic, there is necessarily an increase inthe variation from period to period, of the coefficients A+α_(t) ofequation (16) that correspond to the coefficients c_(k), beyond thevariation that would exist in the quadratic implementation.

[0070] Other aspects of the invention are a computer system programmedto perform any embodiment of the inventive method, and a computerreadable medium which stores code for implementing any embodiment of theinventive method.

[0071] The computer system of FIG. 3 includes processor 1, input device3 coupled to processor 1, and display device 5 coupled to processor 1.Processor 1 is programmed to implement the inventive method in responseto instructions and data entered by user manipulation of input device 3.Computer readable optical disk 7 of FIG. 4 has computer code storedthereon. This code is suitable for programming processor 1 to implementan embodiment of the inventive method.

[0072] Although the invention has been described in connection withspecific preferred embodiments, various modifications of and variationson the described methods and apparatus of the invention will be apparentto those skilled in the art without departing from the scope and spiritof the invention. For example, in variations of the above-describedembodiments, the effects of currency fluctuations in a global portfolioare accounted for.

What is claimed is:
 1. An arithmetic performance attribution method for determining portfolio performance, relative to a benchmark, over multiple time periods t, where t varies from 1 to T, comprising the steps of: (a) determining coefficients c₁=A, and ${c_{2} = \left\lbrack \frac{R - \overset{\_}{R} - {A{\sum\limits_{j\quad t}a_{j\quad t}}}}{\sum\limits_{j\quad t}a_{j\quad t}^{2}} \right\rbrack},$

where A has any predetermined value, a_(jt) is a component of active return, the summation over index j is a summation over all components a_(jt) for period t, ${R = {\left\lbrack {\prod\limits_{t = 1}^{T}\quad \left( {1 + R_{t}} \right)} \right\rbrack - 1}},\quad {\overset{\_}{R} = {\left\lbrack {\prod\limits_{t = 1}^{T}\left( {1 + {\overset{\_}{R}}_{t}} \right)} \right\rbrack - 1}},$

R_(t) is a portfolio return for period t, {overscore (R)}_(t) is a benchmark return for period t, and the components a_(jt) for each period t satisfy ${{\sum\limits_{j}a_{j\quad t}} = {R_{t} - {\overset{\_}{R}}_{t}}};$

and (b) determining the portfolio performance as ${{R - \overset{\_}{R}} = {\sum\limits_{it}\left\lfloor {{c_{1}a_{it}} + {c_{2}a_{it}^{2}}} \right\rfloor}},$

where the summation over index i is a summation over all the terms (c₁a_(it)+C₂a_(it) ²) for period t.
 2. The method of claim 1, wherein A is ${A = {\frac{1}{T}\left\lbrack \frac{\left( {R - \overset{\_}{R}} \right)}{\left( {1 + R} \right)^{1/T} - \left( {1 + \overset{\_}{R}} \right)^{1/T}} \right\rbrack}},$

where R≠{overscore (R)}, or for the special case R={overscore (R)}: A=(1+R)^((T−1)/T).
 3. The method of claim 1, wherein A=1.
 4. An arithmetic performance attribution method for determining portfolio performance, relative to a benchmark, over multiple time periods t, where t varies from 1 to T, comprising the steps of: (a) determining a set of coefficients c_(k), including a coefficient c_(k) for each positive integer k; and (b) determining the portfolio performance as ${{R - \overset{\_}{R}} = {\sum\limits_{i\quad t}{\sum\limits_{k = 1}^{\infty}{c_{k}a_{i\quad t}^{k}}}}},$

where a_(it) is a component of active return for period t, the summation over index i is a summation over all components a_(it) for period t, ${R = {\left\lbrack {\prod\limits_{t = 1}^{T}\left( {1 + R_{t}} \right)} \right\rbrack - 1}},\quad {\overset{\_}{R} = {\left\lbrack {\prod\limits_{t = 1}^{T}\left( {1 + {\overset{\_}{R}}_{t}} \right)} \right\rbrack - 1}},$

R_(t), is a portfolio return for period t, {overscore (R)}_(t) is a benchmark return for period t, and the components a_(it) for each period t satisfy ${{\sum\limits_{i}a_{i\quad t}} = {R_{t} - {\overset{\_}{R}}_{t}}},$

where the summation over index i is a summation over all components a_(it) for said each period t.
 5. The method of claim 4, wherein A is ${A = {\frac{1}{T}\left\lbrack \frac{\left( {R - \overset{\_}{R}} \right)}{\left( {1 + R} \right)^{1/T} - \left( {1 + \overset{\_}{R}} \right)^{1/T}} \right\rbrack}},$

where R≠{overscore (R)}, or for the special case R={overscore (R)}: A=(1+R)^((T−1)/T.)
 6. The method of claim 4, wherein c_(k)=0 for each integer k greater than two, ${c_{1} = A},{c_{2} = \left\lbrack \frac{R - \overset{\_}{R} - {A{\sum\limits_{j\quad t}a_{j\quad t}}}}{\sum\limits_{j\quad t}a_{j\quad t}^{2}} \right\rbrack},$

A has any predetermined value, the summation over index j is a summation over all components a_(jt) for period t, ${R = {\left\lbrack {\prod\limits_{t = 1}^{T}\left( {1 + R_{t}} \right)} \right\rbrack - 1}},\quad {\overset{\_}{R} = {\left\lbrack {\prod\limits_{t = 1}^{T}\left( {1 + {\overset{\_}{R}}_{t}} \right)} \right\rbrack - 1}},$

R_(t) is a portfolio return for period t, {overscore (R)}_(t) is a benchmark return for period t, and the components a_(jt) for each period t satisfy ${\sum\limits_{j}a_{j\quad t}} = {R_{t} - {{\overset{\_}{R}}_{t}.}}$


7. A computer system, comprising: a processor programmed to perform an arithmetic performance attribution computation to determine portfolio performance, relative to a benchmark, over multiple time periods t, where t varies from 1 to T, by determining coefficients c₁=A, and ${c_{2} = \left\lbrack \frac{R - \overset{\_}{R} - {A{\sum\limits_{j\quad t}a_{j\quad t}}}}{\sum\limits_{j\quad t}a_{j\quad t}^{2}} \right\rbrack},$

where A has any predetermined value, a_(jt) is a component of active return, the summation over index j is a summation over all components a_(jt) for period t, R is ${R = {\left\lbrack {\prod\limits_{t = 1}^{T}\left( {1 + R_{t}} \right)} \right\rbrack - 1}},{{\overset{\_}{R}\quad {is}\quad \overset{\_}{R}} = {\left\lbrack {\prod\limits_{t = 1}^{T}\left( {1 + {\overset{\_}{R}}_{t}} \right)} \right\rbrack - 1}},$

R_(t) is a portfolio return for period t, {overscore (R)}_(t) is a benchmark return for period t, and the components a_(jt) for each period t satisfy ${{\sum\limits_{j}a_{j\quad t}} = {R_{t} - {\overset{\_}{R}}_{t}}},$

and determining the portfolio performance as ${{R - \overset{\_}{R}} = {\sum\limits_{i\quad t}\left\lfloor {{c_{1}a_{i\quad t}} + {c_{2}a_{i\quad t}^{2}}} \right\rfloor}},$

where the summation over index i is a summation over all the terms (c₁a_(it)+c_(2 l a) _(it) ²) for period t; and a display device coupled to the processor for displaying a result of the arithmetic performance attribution computation.
 8. The computer system of claim 7, wherein A is ${A = {\frac{1}{T}\left\lbrack \frac{\left( {R - \overset{\_}{R}} \right)}{\left( {1 + R} \right)^{1/T} - \left( {1 + \overset{\_}{R}} \right)^{1/T}} \right\rbrack}},$

where R≠R, or for the special case R={overscore (R)}: A=(1+R)^((T−1)/T).
 9. A computer system, comprising: a processor programmed to perform an arithmetic performance attribution computation to determine portfolio performance, relative to a benchmark, over multiple time periods t, where t varies from 1 to T, by determining a coefficient c_(k) for each integer k greater than zero, and determining the portfolio performance as ${{R - \overset{\_}{R}} = {\sum\limits_{i\quad t}{\sum\limits_{k = 1}^{\infty}{c_{k}a_{i\quad t}^{k}}}}},$

where a_(it) is a component of active return for period t, the summation over index i is a summation over all components a_(it) for period t, ${R = {\left\lbrack {\prod\limits_{t = 1}^{T}\left( {1 + R_{t}} \right)} \right\rbrack - 1}},\quad {\overset{\_}{R} = {\left\lbrack {\prod\limits_{t = 1}^{T}\left( {1 + {\overset{\_}{R}}_{t}} \right)} \right\rbrack - 1}},$

R_(t) is a portfolio return for period t, {overscore (R)}_(t) is a benchmark return for period t, and the components a_(it) for each period t satisfy ${{\sum\limits_{i}a_{i\quad t}} = {R_{t} - {\overset{\_}{R}}_{t}}},$

where the summation over index i is a summation over all components a_(it) for said each period t; and a display device coupled to the processor for displaying a result of the arithmetic performance attribution computation.
 10. The computer system of claim 9, wherein c_(k)=0 for each integer k greater than two, ${c_{1} = A},{c_{2} = \left\lbrack \frac{R - \overset{\_}{R} - {A{\sum\limits_{j\quad t}a_{j\quad t}}}}{\sum\limits_{j\quad t}a_{j\quad t}^{2}} \right\rbrack},$

A has any predetermined value, the summation over index j is a summation over all components a_(jt) for period t, ${R = {\left\lbrack {\prod\limits_{t = 1}^{T}\left( {1 + R_{t}} \right)} \right\rbrack - 1}},\quad {\overset{\_}{R} = {\left\lbrack {\prod\limits_{t = 1}^{T}\left( {1 + {\overset{\_}{R}}_{t}} \right)} \right\rbrack - 1}},$

R_(t) is a portfolio return for period t, {overscore (R)}_(t) is a benchmark return for period t, and the components a_(jt) for each period t satisfy ${\sum\limits_{j}a_{j\quad t}} = {R_{t} - {{\overset{\_}{R}}_{t}.}}$


11. A computer readable medium which stores code for programming a processor to perform an arithmetic performance attribution computation to determine portfolio performance, relative to a benchmark, over multiple time periods t, where t varies from 1 to T, by determining coefficients c₁=A, and ${c_{2} = \left\lbrack \frac{R - \overset{\_}{R} - {A{\sum\limits_{j\quad t}a_{j\quad t}}}}{\sum\limits_{j\quad t}a_{j\quad t}^{2}} \right\rbrack},$

where A has any predetermined value, a_(jt) is a component of active return, the summation over index j is a summation over all components a_(jt) for period t, ${R = {\left\lbrack {\prod\limits_{t = 1}^{T}\left( {1 + R_{t}} \right)} \right\rbrack - 1}},\quad {\overset{\_}{R} = {\left\lbrack {\prod\limits_{t = 1}^{T}\left( {1 + {\overset{\_}{R}}_{t}} \right)} \right\rbrack - 1}},$

is a portfolio return for period t, {overscore (R)}_(t) is a benchmark return for period t, and the components a_(jt) for each period t satisfy ${{\sum\limits_{j}a_{j\quad t}} = {R_{t} - {\overset{\_}{R}}_{t}}},$

and determining the portfolio performance as ${{R - \overset{\_}{R}} = {\sum\limits_{i\quad t}\left\lfloor {{c_{1}a_{i\quad t}} + {c_{2}a_{i\quad t}^{2}}} \right\rfloor}},$

where the summation over index i is a summation over all the terms (c₁a_(it)+c₂a_(it) ²) for period t.
 12. The medium of claim 11, wherein A is ${A = {\frac{1}{T}\left\lbrack \frac{\left( {R - \overset{\_}{R}} \right)}{\left( {1 + R} \right)^{1/T} - \left( {1 + \overset{\_}{R}} \right)^{1/T}} \right\rbrack}},$

where R≠{overscore (R)}, or for the special case R={overscore (R)}: A=(1+R)^((T−1)/T).
 13. A computer readable medium which stores code for programming a processor to perform an arithmetic performance attribution computation to determine portfolio performance, relative to a benchmark, over multiple time periods t, where t varies from 1 to T, by determining a coefficient c_(k) for each integer k greater than zero, and determining the portfolio performance as ${{R - \overset{\_}{R}} = {\sum\limits_{i\quad t}{\sum\limits_{k = 1}^{\infty}{c_{k}a_{i\quad t}^{k}}}}},$

where a_(it) is a component of active return for period t, the summation over index i is a summation over all components a_(it) for period t, ${R = {\left\lbrack {\prod\limits_{t = 1}^{T}\left( {1 + R_{t}} \right)} \right\rbrack - 1}},{\overset{\_}{R} = {\left\lbrack {\prod\limits_{t = 1}^{T}\left( {1 + {\overset{\_}{R}}_{t}} \right)} \right\rbrack - 1}},$

R_(t) is a portfolio return for period t, {overscore (R)}_(t) is a benchmark return for period t, and the components a_(it) for each period t satisfy ${{\sum\limits_{i}a_{it}} = {R_{t} - {\overset{\_}{R}}_{t}}},$

where the summation over index i is a summation over all components a_(it) for said each period t.
 14. The medium of claim 13, wherein c_(k)=0 for each integer k greater than two, ${c_{1} = A},{c_{2} = \left\lbrack \frac{R - \overset{\_}{R} - {A{\sum\limits_{jt}a_{jt}}}}{\sum\limits_{jt}a_{jt}^{2}} \right\rbrack},$

A has any predetermined value, the summation over index j is a summation over components a_(jt) for period t, ${R = {\left\lbrack {\prod\limits_{t = 1}^{T}\left( {1 + R_{t}} \right)} \right\rbrack - 1}},{\overset{\_}{R} = {\left\lbrack {\prod\limits_{t = 1}^{T}\left( {1 + {\overset{\_}{R}}_{t}} \right)} \right\rbrack - 1}},R_{t}$

is a portfolio return for period t, {overscore (R)}_(t) is a benchmark return for period t, and the components a_(jt) for each period t satisfy ${\sum\limits_{j}a_{jt}} = {R_{t} - {\overset{\_}{R}}_{t}}$

where the summation over index j is a summation over all the components a_(jt) for said each period t. 